Vibration control of a nonlinear cantilever beam operating in the 3D space

This paper addresses a control problem of a nonlinear cantilever beam with translating base in the three-dimensional space, wherein the coupled nonlinear dynamics of the transverse, lateral, and longitudinal vibrations of the beam and the base’s motions are considered. The control scheme employs two control inputs applied to the beam’s base to control the base’s position while simultaneously suppressing the beam’s transverse, lateral, and longitudinal vibrations. According to the Hamilton principle, a hybrid model describing the nonlinear coupling dynamics of the beam and the base is established: This model consists of three partial differential equations representing the beam’s dynamics and two ordinary differential equations representing the base’s dynamics. Subsequently, the control laws are designed to move the base to the desired position and attenuate the beam’s vibrations in all three directions. The asymptotic stability of the closed-loop system is proven via the Lyapunov method. Finally, the effectiveness of the designed control scheme is illustrated via the simulation results.

system consisting of a beam and a translating base were determined using the model analysis method. Accordingly, the input shaping control law was designed to position the base and suppress the beam's vibration. Pham et al. 27 used the model parameters obtained through an experiment to design the input shaping control law for a non-uniform beam with a moving base. For the closed-loop control technique, Liu and Chao presented an experimental study on implementing the neuro-fuzzy approach to control a beam-cart system 28 . In their work, the piezoelectric transducers located at the beam's tip were used to suppress the transverse vibration. Another closed-loop control of an Euler-Bernoulli beam with a translating base was done by Shah and Hong 15 , whereas the control of a Timoshenko beam attached to a moving base was presented by Pham et al. 34 .
Most studies on controlling the flexible beam attached to a translating base considered only the linear vibrations 15,[26][27][28] . Under the assumption of small-amplitude vibration, the dynamic tension was ignored. However, in the case of large-amplitude vibrations, the negligence of the dynamic tension (which makes the beam's dynamics nonlinear) can affect the system's performance and stability. Also, the existing studies assumed that the beam's vibration occurred in a plane. Thus, only the transverse or lateral vibrations were considered. Even though Shah and Hong 15 investigate both the transverse and lateral vibrations of an Euler-Bernoulli beam, they ignored the coupled dynamics of the transverse and lateral vibrations; that is, the transverse vibration does not affect the time-evolution of the lateral vibration and vice versa.
Recently, the 3D vibration analysis of a beam has received significant attention [35][36][37][38][39][40] . Do and Pan 38 and Do 39 used the Euler-Bernoulli beam model with large-amplitude vibrations to model a flexible riser. The authors obtained a model describing the system's transverse, lateral, and longitudinal vibrations. He et al. 41 , Ji and Liu 42,43 , and Liu et al. 44 investigated the coupled dynamics of a 3D cantilever beam with a tip payload described by a set of PDEs and ODEs. Also, control problems of 3D beams, wherein the coupled dynamics of nonlinear vibrations, were investigated in the literature. However, these studies have either dealt with a beam attached to a stationary base or proposed control strategies wherein control forces/torques were applied to the beam's tip. The implementation of control actions at the tip is not feasible or practical, see Fig. 1. It might be possible to put an actuator at the tip of a large cantilever beam of a space structure or a riser system. But, for gantry manipulators, surgeon robots, and flexible liquid handling robots, implementing the control forces/torques in the tip is not possible because the tip has to interact with an object or the environment.
The published papers in the literature on controlling cantilever-beam vibrations are restricted to: (i) The cases where the cantilever is affixed on a stationary base, and the free end has 3D motions [41][42][43][44] ; and (ii) the beam is attached to a translating base, but the considered dynamics are linear by ignoring the coupled dynamics between the transversal and lateral vibrations of the beam 15,[26][27][28] . Thus, the control problem of a nonlinear cantilever beam operating in the 3D space without using control input at the tip has not been solved yet.
In this paper, the beam's longitudinal vibration and axial deformation (which makes the beam's dynamics nonlinear) are further considered. In such cases of large-amplitude vibrations, the omission of longitudinal vibration and axial deformation can affect the system's performance and lead to an erroneous result. Henthforth, the control problem of the nonlinear 3D vibrations of a beam affixed on a translating base without any additional actuators is addressed for the first time. The considered system is represented as a gantry robot consisting of the gantry, trolley, and flexible robotic arm depicted in Fig. 1. In the gantry robot, the gantry moves along the k-axis, whereas the trolley moves along the j-axis. A flexible robotic arm with a constant length is fixed to the trolley. The Hamilton principle is used to develop a novel hybrid model describing the nonlinear coupling dynamics of the robotic arm's transverse, lateral, and longitudinal vibrations, and the rigid body motions of the gantry and trolley. Employing the Lyapunov method, boundary control laws are developed for simultaneous control of the trolley's position, the gantry's position, and the robotic arm's 3D vibrations. The asymptotic stability of the closed-loop system is verified. Finally, the simulation results are provided.
The main contributions of this paper are summarized as follows: (i) A novel dynamic model of a flexible beam attached to a translating base, wherein the coupled dynamics of the nonlinear transverse, lateral, and longitudinal vibrations and the base's motions are developed for the first time. (ii) A boundary control strategy using the control forces at the base for simultaneous position control and vibration suppression is designed. (iii) The asymptotic stability of the closed-loop system is proven by using the Lyapunov method, and simulation results are provided.  Figure 1. An example of nonlinear cantilever beams operating in the 3D space: (a) Gantry robot. (www.a-m-c. com/ servo-drives-for-gantry-syste ms), (b) the defined coordinate system and motions.

Problem formulation
In Fig. 1, a flexible robotic arm is modeled as a uniform Euler-Bernoulli beam of length l. The motions of the gantry and trolley are generated by two control forces f z and f y , respectively. The positions of the trolley and gantry are denoted by y(t) and z(t), respectively. The beam's vibrations in the i, j, and k axes are defined as the longitudinal vibration u(x, t), the transverse vibration w(x, t), and the lateral vibration v(x, t), respectively. In this study, the subscripts x and t, i.e., (·) x and (·) t , are the partial derivatives with respect to x and t, respectively, whereas ẏ and ż denotes the total derivative of y(t) and z(t) in t, respectively. The kinetic energy of the entire gantry, trolley, and beam system is given as follows: where ρ and A are the beam's mass density and cross-sectional area; m 1 and m 2 are the gantry's mass and trolley's mass, respectively. The potential energy due to the axial force, the axial deformation, and the bending moment is given as follows: where P(x) = ρA(l − x)g is the axial force generated by the influence of the gravitational acceleration on the beam's elements 45,46 , E denotes Young's modulus, and I y and I z indicate the moments of inertia of the beam. The axial strain ε(x, t) is given by the following approximation 47 : The virtual work done on the system by the boundary control inputs and the friction is given as follows.
where c w , c u , and c v are the structral damping coefficients (i.e., the subscripts w, u, and v stand for transverse, longitudinal, and lateral, respectively). According to Hamilton's principle, the dynamic model of the considered system and the corresponding boundary conditions are obtained as follows.
The dynamics of the considered system are represented by the nonlinear PDE-ODE model in (5)-(12): Eqs. (5)- (10) are PDEs describing the transverse, longitudinal, and lateral vibrations of the robotic arm, respectively, whereas the ODEs in (11) and (12) represent the dynamics of the gantry and the trolley, respectively. Observably, the beam's motion affects the gantry and trolley's motions and vice versa. Additionally, if the potential energy caused by the axial deformation is ignored (i.e., ε 2 = (u x + w 2 x /2 + v 2 x /2) 2 ∼ = 0 ), the nonlinear terms in (5), (7), and (9) vanish. Then, the coupling dynamics between the transverse, lateral, and longitudinal vibrations can be decoupled.

Controller design
The two control objectives are position control and vibration suppression: (i) Move the gantry and trolley carrying the flexible beam to the desired positions, and (ii) suppress the beam's transverse, lateral, and longitudinal vibrations. In this paper, two forces f z and f y applied to the gantry and trolley are used as the control inputs to achieve the control objectives. The position errors of the trolley and gantry are defined as follows: w(0, t) = w x (0, t) = w xx (l, t) = w xxx (l, t) = 0, where K i (i = 1,2,…,6) are the control parameters. The implementation of these control laws requires the measurement of w xxx (0, t) and v xxx (0, t). In practice, these signals can be obtained by using strain gauge sensors attached at the clamped end of the beam.
The following lemmas and assumptions are used for stability analysis of the closed-loop system with the control laws given in (15) and (16).

Assumption 1 21 . The transverse vibration w(x, t), the lateral vibration v(x, t), and the longitudinal vibration u(x, t) of a flexible beam satisfy the following inequalities: u 2
x ≤ w 2 x /2 and u 2 x ≤ v 2 x /2 . By using Lemma 1, we obtain.

Assumption 2 51 . If the potential energy of the system in
Based on the system's mechanical energy, the following Lyapunov function candidate is introduced: where (13) e y = y − y d ,  (13) and (14) converge to zero is guaranteed. Additionally, the control laws are bounded.
Proof of Theorem: Lemma 4 reveals that the Lyapunov function candidate in (23) is a positive definite. According to Lemma 5, we obtain We define the norm of a spatiotemporal function as follows: . Using Lemmas 1 and 4 and Assumption 1, the following inequalities are obtained.
and v t (x, t) 2 based on Lemmas 4 and 5.
Additionally, the following results also imply that w(x, t), u(x, t), and v(x, t) are equicontinuous in t.
Accordingly, we can conclude that lim  (37) reveal that e y , ė y ,e z , and ė z are also bounded. Finally, we can conclude that the control laws in (15) and (16) are bounded. Theorem 1 is proved.

Simulation results
In this section, numerical simulations are performed to illustrate the effectiveness of the proposed control laws. The system parameters used in the numerical simulation are shown in Table 1. According to these system parameters, the control gains in (15) and (16) are selected as K 1 = 750, K 2 = 950, K 3 = 1.12 × 10 4 , K 4 = 550, K 5 = 750, and K 6 = 2.34 × 10 4 . Control parameters K i (i = 1, 2,…, 6) are calculated based on design parameters k i , α n , β j , and δ k (n = 1, 2, 3; j = 1, 2,…, 7; k = 1, 2,…, 9). These design parameters have been selected to satisfy the conditions in (A. 15  The simulations were performed by using MATLAB, wherein the finite difference method was utilized to determine the approximate solutions for the equations of motion. The approximate solutions' accuracy and simulation speed depend on the sizes of the time and space steps (i.e., Δt and Δx, respectively). By using a large time step size, approximate solutions of PDEs are determined quickly. However, a too-large time step size reduces the accuracy of the solution and further leads to instability. Contrarily, the quality of the solutions can be improved by selecting a smaller step size. In this case, the simulation duration increases significantly. Therefore, selecting appropriate step sizes is necessary to guarantee a balance between accuracy and simulation speed. In this paper, the time and space step sizes are selected as follows: Δt = 10 -5 and Δx = 0.075.
The dynamic behavior of the proposed control law (15) and (16) is compared with two typical cases: (i) Using the traditional PD control law and (ii) using the zero-vibration (ZV) input shaping control. For the input shaping control, the ZV input shapers are designed based on the cantilever beam's natural frequencies and damping ratios. The natural frequencies are determined via the solution of the frequency Eq. 26 , whereas the damping ratios are calculated by using the logarithmic decrement algorithm 27 . Figures 2 and 3 illustrate the system's responses under different controllers. Figure 2 shows the trolley's position and gantry's position, whereas Fig. 3 reveal the vibrations of the beam's tip. It shows that the PD controller, input shaping controller, and the proposed controller can move the trolley and gantry to the desired position (i.e., Fig. 2). However, the traditional PD controller cannot deal with the beam's vibrations, see Fig. 3. In this case, vibration suppression was done only based on structural damping; therefore, it requires a significant amount of time. Contrary to the PD control, the system's vibrations under the input shaping control and the proposed control law were quickly suppressed, see Fig. 3. Most tip oscillations were eliminated when the trolley and gantry reached the desired positions (i.e., at t ≈ 4 s). Furthermore, the proposed control law showed an outstanding vibration suppression capability compared with the input shaping control (i.e., see the magnified graphs in    Fig. 6, we consider the system under the influence of disturbances. Two boundary disturbances, d y (t) = 10sin(20πt) and d z (t) = 8sin(20πt), are applied to the trolley and gantry, respectively. As shown in Fig. 6, the proposed control law can still eliminate most of the vibrations of the beam system under boundary disturbance. The sensitivity of the proposed control law to the measurement noises of the sensors is considered in Fig. 7. In this case, 20% noises are added in the feedback signals w xxx (0, t) and v xxx (0, t). Observably, the measurement noises have no significant effects on the responses of the closed-loop system under the proposed control law. The simulation results show that the proposed control law is not too sensitive to disturbances and measurement noises.

Conclusions
This paper investigated a vibration suppression problem of the three-dimensional cantilever beam fixed on a translating base. The equations of motions describing the nonlinear coupling dynamics of the beam's transverse, lateral, longitudinal vibrations, the gantry, and the trolley were developed using the Hamilton principle. Accordingly, the control laws were designed. The asymptotic stability of the closed-loop system in the sense that the beam's transverse vibration, lateral vibration, longitudinal vibration, and gantry's position error and trolley's position error converge to zero was proven via the Lyapunov method. Simulation results showed the effectiveness of the proposed control laws. In practical gantry systems, the length of the robotic arm varies in time, and the system is subjected to disturbances. Our future work will address extending the current control strategy to a varying-length flexible beam with moving base, providing experimental results.

Data availability
The data and codes generated or analyzed in this paper can be available upon the communication with the corresponding author.